Canceling out integral

Question

Bear with my naivety, I wanted to ask if it is possible to cancel out $\int$ with a $\frac{\mathrm d}{\mathrm dx}$. I had $\frac{\partial}{\partial v}$ in a question and I took $\partial v$ to the other side and took integration on both sides. So now on left hand side only $\partial$ remains so will it cancel out with the integral?

Answer 1

Usually when we take integration we remove d or $\partial$

You can say it is cancel out or it is integrated.

Example -

$\frac{d}{dv} x = v^2 + 3$

$dx = (v^2 + 3) dv$

On integrating both sides,

x = $\int(v^2 + 3)dv$

Answer 2

I wouldn't say that $\frac{\mathrm d}{\mathrm dx}$ and $\int$ cancel each other out. Consider the following examples $$\frac{\mathrm d}{\mathrm dx}\int e^t\ \mathrm dt=\frac{\mathrm d}{\mathrm dx}\left(e^t+C\right)=0$$ $$\frac{\mathrm d}{\mathrm dx}\int_0^2 e^x\ \mathrm dx=\frac{\mathrm d}{\mathrm dx}\left(e^2-1\right)=0$$ Study the fundamental theorem of calculus for a deeper understanding of what's going on.

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I'll also address what you mentioned in the comments. Although $\frac{\mathrm dM}{\mathrm dx}$ look and act like a ratio, in modern mathematical analysis, it's not a ratio. It just so happens that $$\mathrm dM=\frac{\mathrm dM}{\mathrm dx}\ \mathrm dx$$ Therefore $$V=\frac{\mathrm dM}{\mathrm dx}$$ $$\int V\ \mathrm dx=\int\frac{\mathrm dM}{\mathrm dx}\ \mathrm dx=\int\ \mathrm dM=M+C$$